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Coulomb's Law

The torsion balance was invented by the Englishman John Mitchell inthe mid 1700's.A torsion balance is a precision instrument which allows for themeasurement of a very small force. A wire is used to hang a leverarm. The wire is uniform in diameter and composition.An object at the end of the lever arm is what is experiencing theforce. The force experienced by the object causes the twisting of thewire. As the wire twists it is possible to measure the angle throughwhich it twists.

The twisting force on a wire is related to the angle through which ittwists by a linear relationship, namely,

Ft*i.t = H 0

where H is a constant of proportionality. By measuring the anglethrough which the wire twists, you can measure the force that is beingexperienced by the object.

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ln some ways the relationship between force and angle for the torsionbalance is similar to the relationship behrveen force and displacementfound in Hookes' Law F = kx, where x is the displacement. ln thecase of the torsion balance, the angle 0 is the comparable todisplacement in Hookes' Law,ln 1785 Charles Coulomb borrowed the idea of the torsion balancefrom the english to do an experiment to determine the relationshipbetween the electrostatic force on charged spheres, and the distancebetween the spheres.

ln his experiment Coulomb took two similarly charged spheres, oneon a fixed rod and a second on the torsion balance-lever arm.

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By touching the two spheres, coulomb was able to ensure the twospheres were equally charged.

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Since like charges repel there is a force pushing the two spheresapart from each other. lt is this repulsive force between similarlycharged spheres which Coulomb measured using his torsion balance.

By turning the knob at the top of the torsion arm he was able tochange the distance between the fixed charged spheres.

How the measurements were taken:

An angular scale near the knob allowed Coulomb tomea$ure the angle through which the wire twisted.

A scale near the spheres allowed Coulomb to measure thedistance between the spheres.

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Coulomb did two experiments: The first was to determine therelationship between the force and the distance, and the second wasto determine how the charges on the two spheres affects the force.

Experiment 1. The Relationship between Force and DistanceConsider a sample experiment with the following results (distanceand angle are in arbitrary measurements):

Distance Angle (0)1.6 0.391.4 0.511.2 0.691.0 1.00.8 1.60.6 2.80.4 6.3

Analvsis:

1) Graph the data with the distance as the independent variableand the angle as the dependent variable.

2) Use logarithms to find the functional dependence betweendistance and angle.3) lf force is proportional to the angle, what is the relationshipbetween the force and the distance?

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Experiment 2: the Relqtionship between Charge and the ForceCoulomb was not interested in an absolute measure of the charge onthe spheres. By touching the two spheres to begin with, Coulombwas able to ensure the two spheres were equally charged.By grounding the sphere on the torsion arm from time to time he wasable to divide the charge several times over so that if he started witha charge Q, on the first run he was comparing the force between twospheres with charge T. Q, and then two spheres with charge yo Q.,and so on ....

Consider a sample experiment with the following results (charge andangle are in arbitrary measurements):

Charge Angle (0)2.4 2.41.0 1.0112 0.50114 0.2s

Analvsis:

1) Graph the data with the charge as the independent variable andthe angle as the dependent variable.

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Coulomb's Law

ln the his first experiment Coulomb determined that the electrostaticforce and the distance between the two charges are related by

Fo< 1ld2The electrostatic force relationshlp is an INVERSE SQUARE LAW.

ln his second experiment Coulomb determined that electrostatic forceand the charge is related by

Fo

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lf two charges each of 1 C are one metre apart they wouldexperience a repulsive force of 8.998 x 10e N. A force of 9 billionNewtons is an enormous force, equal to about 1 million tonnes.

Charges we typically experience are around a microcoulomb (1 p$ =10-6 C) and so we experience electrostatic forces on the order of 1Newton or less.

The charge on an individual electron (e) is - 1.602 x 10-1e C. Thecharge of a proton is equal in magnitude but opposite in sign.

It is a yet to be explained fact that the charge of the electron andproton are equal yet opposite.

1 C of charge is equal to 101e electrons. ln one mole of matter thereare therefore on the order of 105 Coulomb of charge!

lf the charge in matter were not neutral the matter would fly apart witha considerable force.

For instance, when a uranium atom splits during nuclear fission (thenucleus is no longer able to stick together) most of the energy that isreleased from the nuclei is a result of the electrostatic repulsion of the92 protons of positive charge so close together within the nuclei.

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